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We have been working on early algebra. I am trying to teach my son about doing the opposite operation in equations to isolate the variable. He solves by inspection frequently and can often just see things in one and even two step equations. He has been doing what I used to call cross multiplication to solve equations involving fractions or division. He saw the pattern very quickly himself. I never taught him this, he just worked out that was how to get the answer. His view is that he doesn't really need to understand why it works because he has found a method. I get a bit nervous because I wonder whether or not his chosen method (not just in this but in other maths subjects- eg: he multiplies normal numbers algebraically too) will ever disadvantage him in higher maths. It feels to me like he is using a different system for each operation. I try to be as flexible as I can be because I know in maths if you get something one way and people try to teach you an alternative, it becomes very frustrating. I know most kids at schools just learn methods and don't really know or care why they are doing things that way. So what am I really asking? I'm not really sure. Maybe just seeking some reassurance from the creative teachers here that it's okay to go with the flow and also that I won't be disadvantaging him if he only uses cross multiplication, rather than actually understanding what you are doing is an inverse operation.
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